Given a convergent
sequence of contraction mappings, the convergence of the sequence of their fixed
points is investigated in §1 of this paper. The results obtained lead to a
necessary and sufficient condition in order that a separable or a reflexive Banach
space be finite dimensional. An application to differential equations is also
included.
In §2 we consider mappings defined on the cartesian product of two metric spaces
which are contraction mappings in one variable or in each variable separately. Using
some of the results of §1 we prove that, with certain restrictions, such mappings have
fixed point.
